Abstract We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes .
Abstract We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes .
We address the problem of efficiently and effectively compress density operators (DOs), by providing an efficient procedure for learning the most likely DO, given a chosen set of partial information. We explore, in the context of quantum information theory, the generalisation of the maximum entropy estimator for DOs, when the direct dependencies between the subsystems are provided. As a preliminary analysis, we restrict the problem to tripartite systems when two marginals are known. When the marginals are compatible with the existence of a quantum Markov chain (QMC) we show that there exists a recovery procedure for the maximum entropy estimator, and moreover, that for these states many well-known classical results follow. Furthermore, we notice that, contrary to the classical case, two marginals, compatible with some tripartite state, might not be compatible with a QMC. Finally, we provide a new characterisation of quantum conditional independence in light of maximum entropy updating. At this level, all the Hilbert spaces are considered finite dimensional.
Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.
Herein we study the problem of recovering a density operator from a set of compatible marginals, motivated by limitations of physical observations. Given that the set of compatible density operators is not singular, we adopt Jaynes’ principle and wish to characterize a compatible density operator with maximum entropy. We first show that comparing the entropy of compatible density operators is complete for the quantum computational complexity class QSZK, even for the simplest case of 3-chains. Then, we focus on the particular case of quantum Markov chains and trees and establish that for these cases, there exists a procedure polynomial in the number of subsystems that constructs the maximum entropy compatible density operator. Moreover, we extend the Chow–Liu algorithm to the same subclass of quantum states.
The statistics of local measurements of joint quantum systems can sometimes be used to distinguish the spatiotemporal structure in which they were measured. We first prove that every bipartite separable density matrix is temporally compatible with direct causal influence for arbitrary finite-dimensional quantum systems and measurements of a tomographically complete class of observables, which includes all Pauli observables in the case of multi-qubit systems. Equivalently, if a bipartite density matrix is not temporally compatible with direct causal influence, then it must be entangled. We also provide an operational meaning for the two temporal evolutions consistent with such correlations in terms of generalized dephasing channels and pretty good measurements. The two temporal evolutions are Bayesian inverses of each other, which is different from them being Petz recovery maps of each other. Finally, we prove necessary and sufficient conditions for an arbitrary bipartite quantum state to be temporally compatible, thereby providing a temporal analogue of the positive partial transpose criterion valid for quantum systems of any dimension.
We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional tensor-product Hilbert spaces. The intrinsic construction starts from a joint state and its conditional-independence properties. The extrinsic construction assembles a state sequentially from prescribed local quantum kernels, following an ordering compatible with the arrows of the DAG. For the intrinsic construction, we prove the equivalence of the ordered, local, and global directed Markov properties, together with entropy, recursive-factorization, and logarithmic characterizations. The extrinsic construction always gives a normalized state and recovers each kernel as a conditional on all preceding systems. The same kernel, however, need not be recovered from the marginal on the vertex and its parents; a three-qubit example exhibits this obstruction. We prove that independence of the chosen topological ordering is sufficient exactly for transitive DAGs: every order-invariant kernel family then yields an intrinsically directed Markov state. Finally, we associate a logarithmic candidate with every positive definite state and DAG, prove that it is subnormalized, and show that the trace-one candidate is a directed Markov state. Both the candidate and the excess global information are invariant under DAG Markov equivalence.
We study when local reduced density operators, viewed as quantum marginals, can be assembled into a global quantum state with a prescribed Markov structure. The starting point is a canonical logarithmic construction T(R), the noncommutative analogue of the junction-tree formula for decomposable graphical models. Unlike in the classical case, this formal construction may fail: noncommutativity can prevent it from being a normalized state with the prescribed marginals. We prove that this obstruction is captured exactly by a trace condition. For two overlapping marginals, and for clique marginals on a chordal graph, the condition Tr(T(R))=1 is equivalent to the existence of a quantum Markov completion. When it exists, the completion is unique, equal to T(R), and selected by the maximum entropy principle. In the two-clique case, we also give an equivalent conditional reconstruction characterization: the two natural one-sided sandwich reconstructions agree if and only if the trace condition holds. We introduce the global quantum information gI(G)ρ associated with a chordal graph G and show that it is a relative-entropy discrepancy from ρ to the logarithmic candidate, with a trace correction when the candidate is not normalized. We also prove an intersection property for strictly positive quantum conditional independence. Three-qubit Pauli examples illustrate how the quantum obstructions are real: local consistency, feasibility, Markov feasibility, and maximum entropy can all separate.